 Failure of finite-round game equivalence in infinite languages

For countably many unary predicates $P_i$, let $M$ contain infinitely many copies of each finite membership pattern and let $N$ add an element satisfying all predicates. Every finite vocabulary reduct has infinitely many elements in every Boolean cell in both structures, so $M\equiv N$. The Spoiler nevertheless wins the one-round full-language game by selecting the new element, whose complete atomic pattern has no counterpart in $M$.